Question

Find a recursive formula for the sequence:

Answer 1

Step-by-step explanation:

$a _{n} = - 2a _{n - 1} + 1$

Answer 2

Answer:

  $a_n=-2a_{n-1}+1$

Step-by-step explanation:

Successive terms do not have a common difference or ratio, so the sequence is not arithmetic or geometric. We can get a clue as to its nature a couple of places:

1. from the alternating signs, says there is a negative multiplier involved
2. from the answer choices

It is helpful to look at the differences of terms:

  -5-3 = -8

  11 -(-5) = 16

  -21-11 = -32

These have a common ratio of -2, so match an exponential form with a ratio of -2 (the last answer choice). Trying that choice with n=1, 2, 3, 4 gives ...

  a[1] = 3

  a[2] = -2(3) +1 = -5

  a[3] = -2(-5) +1 = 11

  a[4] = -2(11) +1 = -21

These match the given sequence, so the appropriate recursive formula is ...

  $\boxed{a_n=-2a_{n-1}+1}$